The short answer: The brain is a huge set of neurons that produce electrical activity as part of their function. We can sense this electrical activity and correlate it with brain states (alert, asleep, awake, meditating, etc). The passing of the brain between states (e.g. alert to asleep) is marked by modifications to the patterns of electrical activity.
The not so short answer: If you are "serious" about processing EEGs, get yourself a copy of Niedermeyer and Da Silva's: Electroencephalography: Basic principles, clinical applications and related fields. It is a monumental book on EEG and will answer all of your questions in excruciating detail. But, you have to give the book the time it needs. Especially sections 2,3,4,5 on the basics of signal generation and acquisition and then 11,12,18,26,30,31 and 56 which talk about modifications caused by certain diseases and during sleep.
Now, the brain is a lump of neurons. Different types of neurons that are there for different types of tasks. Some of them respond to electrical activity (from the nervous system) by propagating or inhibiting electrical activity. We can see this in the lab, by observing how does one single neuron behaves but to date it has been impossible to observe what EACH neuron is doing in-vivo in a functioning brain. This is the "promise" of Neuroimaging.
So, when it comes to electroencephalography, an electrical activity (of the brain) is sensed by an instrument (the EEG aparratus). Actually, in this case, a ridiculously small number of electrodes (a few tenths) is used to sense the combined functioning of a ridiculously large number of neurons (a few billions).
Because of this "imbalance" we see patterns in the electroencephalography measurements that, macroscopically, correspond to brain function. We don't really know what is happening at the neuron level, but we can observe waves of electrical activity spreading throughout the surface of the brain and we hypothesize that this is probably because of the brain's properties of segregation and specialisation (and this, go through the introduction, at least once).
In other words, certain parts of the brain are devoted to certain functions and they "connect" together to exchange information as the brain functions.
Because of this, certain "states" of cognition can be cross referenced with certain electrical patterns.
If you ask an average someone to close their eyes and relax, their brain will kick into "alpha" wave activity. If it doesn't, then this might suggest that this average someone, is not so average after all.
Theta waves are associated with hippocampal activity. If that activity is found to deviate from what is considered "normal", then this is a sign for things starting to go wrong (please see section 3.2 but also 3.1 for some general remarks.). Theta waves are also associated with another "big mystery" of brain function called "sleep".
Spindles are associated with sleep activity and are correlated with phases of sleep. Lack of spindles, is usually not a good sign for brain function.
As far as "...drop of voltage/waves", "...induces.." are concerned: The waves denote brain states. As you emerge from sleep, the brain makes a transition. So, it is expected that you will stop exhibiting theta waves and sleep spindles and you will come back to having alpha waves and ultimately normal dynamic brain activity. Anything that mentions drop of this wave and increase on that wave, etc, denotes the passing of the brain through different states.
- As a side note to this, you probably are aware of various youtube videos claiming to relax, boost creativity, put you to sleep and other claims. I am not sure which and how many of them work but their "basis" is that you can synchronise the brain with an external signal source and induce some sort of brain wave to it. So, this is like saying "well, if the brain produces theta waves when it goes to sleep, why don't we replay some sound source, modulated at theta wave frequencies, to the brain to get it to synchronise and go to sleep?". In fact, certain practices such as meditation are like conscious attempts of a brain's "user" to put it into certain regenerative states. But which of these things work, inferred scientifically, is currently an open question.
Hope this helps.