I am reading verneer's "building a bankroll". I can understand most of it but just don't get how to calculate implied odds in these situation. It will be a big favor if someone illuminate..

In this instance he means the ratio of the remaining stack to the amount you need to put in.

In the example villain has roughly 10 times the money in his stack compared to the $2,25 you need to call hence the 9:1.

Thank you so much. That's easy

The logic is that, even thought you're only a 7.5:1 dog to flop a set, when you do flop a set, you will not stack this opponent anywhere near often enough for calling here to be +EV.

In general, calling a raise with a small pair you'd like at a minimum 15:1 to 20:1 to set mine. That number varies based on how loose/tight villain is and how high your pocket pair is (e.g., 88 wins unimproved more often than 33).

Not only do you need to calculate implied (speculative) chips you may gain by the end of the hand you need to be up against an opponent who will 'most likely' put them in there for you to collect.

Outside of set mining you can have a draw on the Flop ... say just a flush draw. Even if you 'only' need 3 to 1 to continue (or actually may be a statitical favorite if you have other outs) you 'know' that this V wont pay you off if any of your cards come on the Turn or River. So you look at the V chip stack and raise/overbet to the point that the V will have to put in enough chips to make it worth your while to see a showdown. In a lot of cases the V will fold and you pick up the pot with the worst hand unimproved. The more you can get an opponent to fold in this spot, the less you actually need to make your hand when a V does call.

You are using the chips you want (that would normally be 'implied') to create FE (fold equity) for the V to get away from his hand since he doesn't want to pay off your draw .. even before it happens!!

This is not a high volume move, but the point is that without the 'implied' odds that the chip stacks create (whether they are in the pot or not) they need to 'be there' in order to take either action (call or raise).

If you do have a V that is likely to pay you off, then you might as well not risk too many chips and just wait until you make your hand to put them in the middle. GL

Is this a rationale behind check-raising the flop with a draw? You call a $0.15 raise preflop with J T. Flop comes K 2 3. V cbets $0.20, but is unlikely to put more money in if a comes on the turn. So raising to $0.45 essentially gets you the bigger pot on the turn if he calls and your draw hits, but also has the added benefit of possibly getting the best hand to fold.

So if he folds x% of the time, but you know he has a better K the other times and won't fold, and you will fold if the turn is a non-heart (obviously not always true, but ignoring that for now):

0=x*$0.50+[(1-x)*(.80*$-0.45+.20*$1.20)] (rounding to 20% for your flush draw hitting the turn)
0=.5x+[(1-x)*(-0.12)]
x=.19

So you can "realize" your implied odds without even hitting your draw if he folds 19% of the time to your raise?

Your highlighting game on point

In a word, yes. Check-raising with a draw is generally "good practice", because it's hard to get paid off when you hit, because a villain will often check back when the draw comes in, but he might just fold to the raise, so you don't have to suck out.
In some spots, check-calling with at least some of your draws is fine, but it's usually the weaker players (aka calling stations) that routinely get taken to valuetown when they play draws passively OOP.