Why horse races?

I wondered that too.

They do also use the lottery. In Ep.1,
Gower: Well, I guess you heard [Philip Pearson] won the lottery yesterday, huh?
MacLaren: You mean you had to drop the charges?
Gower: No, I mean that he won the freaking lottery! Six numbers out of seven. 92-grand and change.
Maclaren: Lucky kid.
Gower: Not if I catch him with the bag of smack I'm pretty sure he's walking around with.

I guess if you're in jail, it's easier to buy a lottery ticket than to bet on a race. Plus, he started off with nothing but the money in his wallet, so to match that, he'd have to make a series of bets, rolling over his winnings, which would probably look more suspicious than getting lucky once.

Keep in they are probably betting through bookies which has the advantage of being even less conspicuous than going to the track. Besides that;

1) Big wins mean big taxes
2) A bunch of small hits are less conspicuous. Big winners gets lots of press which a traveler does not want.
3) They are trying to minimize the changes to history and that what their method does somewhat better than the alternative.

Betting can be done through bookies. Transactions are all in cash, off the grid and untraceable. And with a series of smaller scores rather than one big jackpot no eyebrows are ever raised.

I think it's how they get money to fund their "missions", so I assumed they have permission to do so.

Stocks you need a social security number.

Horse racing is a cash investment, very liquid and there is a big black market to find other bettors.

They did the local lottery in the first episode. 6 out of 7 numbers correct, so it wouldn't be so much money. That was what was used to get the cars and the garage/lair and all of the computers. I think knowing horse races is easier to remember, since it is names, and it isn't as popular to the general public as to whom the winners are. Some of the odds can be crazy high, if the horse wins.

As others have said, it's to keep a low profile. Winning some big lotto for millions would be too attention grabbing.

Also, Phillip most always uses someone else to do the betting for him. I think I saw him ask two different guys to bet on horses for him... and then also gave winning horses to other people in need. I think he never actually bet on them himself, or probably did it one time only... It would raise suspicion if one person kept going in there randomly and winning a lot.