Does the Player's Handbook contain anything that would specifically help dungeon masters?
I am curious about this because I am trying to find out if I can honestly learn a lot about Dungeons and Dragons by buying the book.
Yes, the Player’s Handbook contains the majority of the actual game rules, which a DM is required to know to run the game. The DM needs to read all three core books — Player’s Handbook, Dungeon Master’s Guide, and Monster Manual — to be able to run the game using those books. (I say “those books” because the core 3 is not the only source to choose from; see below.)
That is assuming that you’re talking about learning to play and run the game using the core books. If you’re using the free Basic Rules, you don’t need to read the PHB, but the same idea applies: you need to read the Player’s Basic Rules before the DM’s Basic Rules will make sense.
If you’re using the Starter Set, then that contains everything you need and you don’t need the core books or the Basic Rules. In that case though, you still need to read everything the box contains.
The introduction of the Dungeon Master Guide (DMG) states
This book has two important companions: the Player's Handbook, which contains the rules your players need to create characters and the rules you need to run the game, and the Monster Manual, which contains ready-touse monsters to populate your D&D world.
The Player Handbook (PHB from now on) describes most of the rules in the game. The use of abilities, price of equipment, rules for spellcasting and the rules for combat are all written in the PHB, and the DM has to know most of these.
The DM should also be familiar with the characters' abilities in order to be able to provide sensible challenges (i.e. not killing the whole party or making it too easy).
If you don't want to spend money without knowing that you will keep playing, WotC provides free Basic Rules, which contain the... basic rules. It doesn't have all the classes and races from PHB, but does the job in the ruling part. But again: if your players are using things from PHB, you should be familiar with it.
To look at it a bit differently: The DM runs monsters--and they operate in the same world as the players. Thus anything the players use to interact with the world the DM will need to know for his monsters to function in the world.
As far as I'm concerned it should be called the Core Rulebook rather than Player's Handbook.