1. Why do we bet?

To begin with, we must first quickly discuss the concept of opportunity cost. This is a concept that has leaked somewhat into every day life, but is still vastly underused. A quick google search defines opportunity cost as "the loss of potential gain from other alternatives when one alternative is chosen." This applies in a variety of areas in poker, including many that are not topical here: game selection (and specifically, when not to play a game), specialization (tournaments vs cash? NLH vs PLO vs mixed games?), seat selection (you generally want position on the players that are most likely to leak money to you, whether that simply be because they play the most hands or their games are extremely exploitable OOP vs IP, but sometimes, you might want to be OOP on the shortstack bc nobody else at the table realizes that they're aggressively jamming with their short-stack so you can use them to reopen the action), etc.

Getting back to the titular question, "Why do we bet?": This is a question frequently asked and discussed in poker material, and to my knowledge, while all answers are correct at least from some angles, they usually overfit for specific scenarios and thus are too complicated and aren't comprehensive enough to capture the entirety of betting. Really, the only reason needed to explain why we bet is that we bet to capture a larger % of the pot than we would if we were to check (in other words, the EV of betting is higher than its opportunity cost--the highest EV of all its other possible actions). That's it. And starting with this baseline helps explain basically every individual concept of poker theory (the impact of position, overbets, down bets, donk bets, range checks, check-raising, etc).

What this very simple fundamental understanding of betting in poker then helps us realize is that by definition, no betting strategy in equilibrium for one's entire range can yield an EV larger than the size of the pot at any given point, because at any node in which a given set of strategies leads to this result, the opponent can just do better by folding everything. This is not to say that no individual hand can have an EV that is larger than size of the pot; we are referring to the strategy for all hands in one's range as a whole. In addition, no individual hand should be bet that if by betting, the hand makes a smaller % of the pot at equilibrium than it would had it checked (which is itself usually a clearly positive pot share).

Furthermore, this understanding for why we bet can also explain a lot of the common answers for that question. "We bet to extract value with our best hands" = we bet because the EV of betting is higher than the EV of checking for our best hands (which is very positive in this case). "We bet to win pots with hands that otherwise wouldn't win at showdown" = we bet because the EV of betting is higher than the EV of checking for our bluffs (which is close to zero in this case). "We bet to 'charge' opponents for trying to realize the equity of their hands" = we bet because the EV of betting is higher than the EV of checking (as we're forcing the opponent to put more money in the pot with hands that only rarely win).