The Moments screen can be found by clicking on 'team,' 'upgrades,' and 'moments.' Once you buy/open some Moments boxes, equipping them will be self explanatory when you're at the Moments screen. They're the most powerful tool that tsb has ever introduced. They add a layer of strategy and when multiple moments are in play, can be a fun wrinkle in an otherwise cookie-cutter TSB world.

Moments range from level 1 to level 5. The higher the level, the higher the boost. Each time you pull a duplicate Moment, you get 'points' towards the next Moments level. Once you hit the various thresholds for the next level, you can level them with blue cash at the Moments screen. If your Moments are already level 5 and you get duplicates, you'll receive a token amount of blue cash or gold.
Here are some Moments tidbits:
-Moments that help with skills(most of the 'basic'/bronze moments) stack with skills. The Cleveland Indians gives 30% to WoH for any Cleveland batter. My Kenny Lofton has skills at 30/30. With the Cleveland Moments equipped, my boost will be: 1.3(1.3)(1.3)=119.7%. If an xfactor is equipped, it would also stack accordingly.
-Moments that help with club points will boost points for your entire team. There is a 'moments multiplier' section that stacks with all other multipliers(tiers, accolades/events), which can create huge scores. In the example below, I have six players(Taylor and five pitchers) that triggered a 50% Moments boost to the Moments multiplier. 1.5(50% boost) to the 6th power(number of players that triggered the boost)=11.39 towards my Moments multiplier.

-In order for the club bonus moments to trigger, the player must score bonus points. One of the first things I look at is the team bonus for the event. If the Braves are a team RBI bonus, for example, I know that all the Braves righty OFers(Hank Moments) will trigger their moments boost. This makes otherwise useless players worth keeping. Obviously, if the player themselves are a bonus, I look for a corresponding Moments boost.
-Moments can be very specific so it's important to read the details. If a Moments is for a 1b Lefty on the Yankees(Gehrig Moments), for example, then only 1b lefty Yankees that score bonus points will trigger the Moments boost.
-Switch hitters will work with righty specific batters, but NOT with lefty specific bats. For example, the Max3k moments are for righty bats and switch hitting Dodgers will work(as well as rightys, of course). The Babe moments are for lefty OF, but swich hitting OFs will NOT work.
-ML(created legend) will work with moments. Simply change your favorite team accordingly.
-New moments are generally introduced on Fridays and most will get shuffled in the 'regular' moments box with previously released moments.
-A few non VIPs have reportedly been jailed for scoring 2+bill points/game. Hope to get this sorted soon, but worth sharing for the time being.
If you have any questions, I'll be happy to answer them. Please keep all comments specific to moments. If you want to vent about 'xyz,' we have other threads.
Moments Teams/Skills:
Basic:
Angels, Dbacks=vs lhp(caps at 22%)
A's, Marlins, Reds=tourney games(caps at 30%)
Astros, Cubs=1st pitch(caps at 24%)
Blue Jays=club points when bonus pts are scored(caps at 22%); think of it like a pick'em bonus
Braves, Giants, Yankees=RISP(caps at 22%)
Brewers, Nats=prime games(caps at 24%)
Cards, Rockies, Wsox=close game(caps at 24%
Dodgers=bonus games(caps at 30%)
Indians, Mariners, Twins=WoH(caps at 30%)
Mets, Rsox, Royals =late innings(caps at 24%);works in woh
Orioles, Rangers=2 outs(caps at 24%)
Padres, Phillies=cvc(caps at 30%)
Pirates, Rays, Tigers=first 3 innings(caps at 30%); also works in cvc
Comments
I wish they would allow you the option to buy extra “slots” to use additional moments.
I don't think it is a coincidence that Friday was the first weekend that I went after Moments, for Ruth and Gehrig.
I had a whopping 1.44 Moments Multiplier for them!!!
And currently jailed. Not cool.
1.1 to the 5th power(your 5 det bats)(1.4-jackie)(1.5-babe)=3.38