QPSK in its raw form would appear just as the OP has shown, where the information is encoded into four symbols given by the four phase states alone. The problem is this has a very wide transmit spectrum given the pulse shapes are rectangular and the Fourier Transform of a rectangular pulse is a Sinc function (whose peaks roll-off in frequency only at the rate of 1/f). Wireless spectrum is an expensive commodity, so we go through great lengths in terrestrial wireless communications to limit spectral occupancy and in the process improve spectral efficiency. This is the only reason for pulse shaping, but it is significant. By more slowly transitioning between symbols rather than the abrupt transitions shown, we greatly limit the amount of spectrum needed (the spectrum will be given by the Fourier Transform of the pulse shape). Effective pulse shaping does this while controlling the inter-symbol interference that is added in the process.
The graphic below demonstrates this for a 16QAM waveform where the real portion of the time domain waveform is shown in the upper part of the graphic with and without pulse shaping. The lower part of the graphic shows comparative spectrum where we see the significant reduction in spectral occupancy due to the pulse shaping provided.
I see this graphic is the same one as what the OP linked, so to further explain: This is how the envelope of the signal would look like in the air (we don't see the actual carrier frequency in this graphic). This graphic was for QAM so does have information in the amplitude as well as phase, but for QPSK case the amplitude would carry no information. But even with the QAM case, the pulse shaping modifies the amplitude ONLY to reduce the spectrum, and modifies it in such a way to control the trajectory between our symbol samples of interest, while still passing through those samples defined by the modulation exactly at the proper symbol timing locations.
Below is an eye diagram showing the real and complex waveform for the QPSK case (The eye-diagram similar to the time domain waveform shown above, but repeating synchronous to symbol boundaries)
Show this shows how the over the air waveform would look (in actuality we typically do half the pulse shaping in the transmitter with a root-raised cosine filter while this plot shows the result after the second root-raised cosine filter in the receiver), but in general we are seeing the amplitude and phase of the time domain waveform for all samples, and the job of the receiver is to determine the sampling locations where the red dots are shown (the center of each symbol), and we then determine the magnitude and phase of those to demodulate (or demap) the QPSK data.
To see what is transmitted "over the air", the sketch below begins to demonstrate how the envelope of the carrier would be modulated by the pulse shaping, but doesn't provide the whole picture as the phase would also be transitioning from symbol to symbol so would not align with what would occur if we continued to transmit one symbol repeatedly. I added more details further below showing the exact waveforms expected. These are the real waveforms that are transmitted at the antenna rather than the baseband equivalent waveforms showing the complex envelope typically with two plots as I (real) and Q (imaginary). The sinusoidal carrier could be any frequency and the envelope would not change, including frequency = 0 which would then represent the eye diagram and I/Q plots that I have shown above (and must then be a complex waveform to represent).
A further note is the OP used symbol mapping that is not optimally gray coded (the 180 degree transition depicted in the diagram going from symbol 00 to 10 should be reserved for transitions where both bits change such that only one bit changes for all symbols that are close to each other).
Also note how this is similar to the operation of windowing prior to computing the FFT to reduce spectral leakage.
To avoid any confusion based on subsequent questions in the comments, here is the actual results on what the OP's QPSK waveform would look like at "RF" using properly gray-coded symbols using the following mapping of symbol to constellation:
We will start with the final waveform in the receiver as the desired QPSK constellation and work back to what is transmitted over the air by adding in the carrier offset and seeing the result:
First the I and Q waveform at baseband for a QPSK waveform pulse-shaped with a raised cosine filter with alpha = 0.3 would appear as the following in the receiver once all the carrier offsets are removed:
The constellation of the above is formed by plotting I vs Q on the complex plane. These trajectories represent the phase and amplitude of our RF carrier at every moment in time, and here I show the one instance that occurs at every integer symbol number in red, which is the only point over the duration of each symbol where the phase and magnitudes of the carrier will actually represent what we expect for QPSK (every other sample in between will be in transition at other phases and amplitudes!).
With an actual carrier frequency added, this same sequence would become the waveform shown in the plot below, where the amplitude of the envelope will match the amplitude of the constellation in the plot above:
Zooming in on the first 10 samples, this is what the OP would see for the sequence given (this is for the sequence 0,2,1,3,0,2,1,3,1,1,2,2.... specifically). The symbol sample points are aligned in this plot with the integer symbol locations on the horizontal axis. This was done with a carrier frequency that is 10 times the symbol rate, but if we increased the carrier frequency further this plot showing the envelope would just get solid but appear identically the same otherwise.
And the same thing as we reduce the carrier to 4 times the symbol rate, showing that we will see the same basic envelope shape regardless of which carrier frequency is used (and as noted if the carrier gets too small compared to the symbol rate we need to use a complex signal to properly represent the waveform, exactly as we do in transceiver implementations):
Note that these are all done with a raised cosine filtered pulse-shape, but what is typically transmitted over the air is root-raised cosine filtered. To be complete, below is the plot showing the same waveform with a root-raised cosine pulse shape:
With close observation of the above plot against the unmodulated carrier aligned with the first symbol we see how the modulated carrier is continuously changing in phase during the course of each symbol and, as also made clear by the constellation diagram of the modulated waveform, is only at the correct amplitude and phase according to the QPSK constellation only over very short durations, and in this specific case after being passed through a second RRC filter. (If the pulse-shape is altogether eliminated then the waveform will stay at the correct amplitude and phase for the entire symbol, and as the pulse shape roll-off factor is increased, this will approach that).
The above plots and descriptions used Raised Cosine (and Root-Raised Cosine) pulse shaping but an alternate harris-Moerder filter is described in this paper http://eon.sdsu.edu/~seshagir/SDR05.pdf. Using this as the pulse shaping filter results in an order of magnitude lower EVM for the same number of taps. This isn't commonly used even today (15 years later) to my knowledge since the root-raised cosine filter is baked into our specifications but certainly is very compelling to be aware of for consideration in future radios (I added a question to see if there is further experience with its use: Use of the harris-Moerder Nyquist Pulse Shaping Filter which may have additional information on this.)