1. Why grappling?
Ever wanted to armlock an ogre? Drag that cowardly, bow-sniping Ranger over a cliff? Hold down two orcs while kicking the third? Grappling can help you do all this and more.
Why grapple? Easy; Grappling is mechanically unfair. To start, most enemies in D&D derive their strengths from unrestricted movement, free reign to cast spells, ability to target certain PCs in the party, high ACs, etc. All this is totally shut down by a good grappler. Although it looks like an innocuous condition, being "Grappled" (and/or Prone/Restrained on top of that) is a serious hindrance for most enemies. If you can impose these conditions at will, you will dominate the battlefield.
When thinking about mechanics, I like to think about their axis of interaction. Attack rolls are opposed by AC. Spells are opposed by saving throws. High damage is opposed by high HP. Looking over the Monster Manual, most enemies are generally well prepared to fight attack rolls/spells/damage, by virtue of high AC, high saves, and/or high HP. Not every monster has those traits (bless your soul, little goblin), but over the course of your travels, you will often encounter monsters that directly counter these angles of interaction.
Then there's grappling. Grappling is a skill contest (not an attack roll!) based on the Athletics skill. Most monsters, even those with high strength scores, have the Athletics/Acrobatics checks of a 1st level commoner. Sure, you will fight monsters that can't get grappled (Ghosts...why did it have to be ghosts...), but the vast majority of monsters will be crying uncle against a good grappler. That lets you dictate how the enemy moves, who they attack, how they take hits, etc. That is a degree of battlefield control most classes can't boast.
Overall, here are the pros and cons of grappling in 5.0.
Grappling Strengths
Very hard to interact with. Most monsters are about as good at Athletics/Acrobatics as a first level character.
Negates mobility. It's amazing how many opponents rely on movement and how many can't do anything once that speed hits 0.
Makes monsters easier for your party to hit. Grapple them, knock them prone, don't let them stand up, don't let the run around: it's "I hold, you punch" at its finest.
Cripples offensive capabilities. Grappled monsters already don't get free reign to choose targets or use their weapons. Now give them disadvantage on top of that, or even take their weapons away.
Combo potential; Grapple a spellcaster in a zone of Silence and no one will hear him tap. Hold two guys in place while your Wizard rains Fireball on their heads. Plant the badguy on the ground as your rogue stabs away. Jump out of a window while holding your targets. The possibilities are endless!
Grappling Weaknesses
Weak against large numbers of opponents. As in real life (for those martial artists in the audience), grappling is not particularly effective against multiple attackers. You can only meaningfully engage with as many attackers as you have hands, which is often just going to be two.
Relatively low damage. Most grappling builds don't deal too much damage on their own, and will have to rely on either allies for massive damage output.
Low mobility. You will often find yourself in situations where you can't dart around the battlefield without releasing your grappled targets back into the fight. This can be a problem in chaotic battles.
When you get shut down, you get shut down HARD. Come up against a monster that's immune to shove and grapple? Have fun poking him for the rest of the encounter. Run into that one spellcaster who pre-cast Freedom of Movement? Better hope someone has Dispel Magic. Dragon won't land from the sky? Grow wings or throw insults at him.
Not for the faint of heart: Don't all guides have some ridiculous weakness like this? Seriously though; if you don't want to get in the face of something two size categories larger than you, a frenzied enemy with a greatsword, or a wizard casting damage-maximized evocation spells, this isn't the class for you.
http://www.enworld.org/forum/showthread.php?468737-The-Grappler-s-Manual-%282-0%29-Grappling-in-5th-Edition