What are the benefits of waiting to drop troops second?

Most of the strategy in Clash Royale is about gaining an "elixir advantage" on your opponent. For instance, if your opponent plays a 4 elixir card and you counter with a 3 elixir card that stops it completely, you have just gained a 1 elixir advantage.

Stacking up these "mistakes" your opponent makes is key to victory. These advantages can be used later in the round to drop a sizeable force that your opponent can't counter, and lets you push up to the tower and inflict damage.

This means that for a lot of players, their deck is reactionary. They want to see what you play so that they can play the counter card and earn an advantage.

Waiting can work against a player, though. I will generally let my elixir stack up to 10, but then at 10 I'll play something. If my opponent doesn't play anything and I climb back to 10 again, I've just gained an elixir advantage for free.

Dropping a defensive tower may seem like the obvious choice, but defensive structures take damage over time, so their effectiveness is less the longer the round drags on without your enemy playing a card. I will tend to play something soft and cheap, like spear goblins, and then back it up with another card if it seems like I could push towards their tower in a lane.

I've had a match where I waited the entire time for the other person to go. The game ended in a draw with anyone dropping a troop. The reason I waited so long, was because I had a bad initial draw for an aggressive move. So whether you should wait or not depends on the cards you get and the deck you are running. If you are running a low cost deck, a tower deck, or a spawner deck, a lot of the time you will want to play earlier. However, higher cost decks tend to want to go second, because they want to see where it is safe to deploy troops, and what they need to use to counter, to maximize elixir efficiency.