Proficiency Bonus additions and HD increases
On pages 5 and 6 of the article from UA, you'll find a change in how to apply proficiency bonuses and HD advancement.
Once the creature (for example, a wolf) becomes your animal companion its proficiency bonuses are tied to yours. You no longer use the ones on p. 8 of the Monster Manual.
When you are 3rd level, its bonus matches yours at +2.
When you are 5th level, it's proficiency bonus matches yours at +3
AC 13 (natural armor)
HP 11 (2d8 +2)
STR 12(+1) DEX 15(+2) CON 12 (+1) INT 3(−4) WIS 12 (+1) CHA 6 (−2)
The animal companion's proficiency bonus is applied to (this is sometimes significant):
The new damage roll for the wolf's bite is 2d4+2+2 at 3rd level; 2d4+2+3 at 5th level; 2d4+2+4 at 9th level, and so on. While early on that + to damage is nice, having +6 damage at 18th level won't make anyone swoon.
It also becomes proficient with all saving throws.
That's a nice feature. Every saving throw the animal companion has to make gets the proficiency bonus (on top of any ability based bonus).
Your animal companion gains proficiency in two skills of your choice. Whether or not these replace any proficiency or unusual skill from the MM, or adds to them, isn't specified. (These are play test rules, so maybe try it both ways at your table). Pact Tactics and Keen Senses (hearing and smell) would be fine to stay, but that's a DM's call.
For each level you gain after 3rd, your animal companion gains an additional hit die and increases its hit points accordingly.
If you get to 11th level, you'll have added a hit die at each level 4-11. (11d10+con bonus) Your animal companion will add a die at each level up: the wolf would have 10d8(+con bonus) at that point.
ASI based ability increases
Leveling up the animal companion is tied to your Ranger level. Bumping the Dex to 16 at 4th level will help attack and damage a bit. It's up to you where to put the points.
Whenever you gain the Ability Score Improvement class feature, your
companion’s abilities also improve. Your companion can increase one ability score of your choice by 2, or it can increase two ability scores of your choice by 1. (Nothing can go over 20)
If you play a straight ranger, you should see ASI increases at 4,8, etc. Do the same for your animal companion. It can't take feats, as you may be able to, but you can boost its stats just as you'd boost your own. If you are multi-classing, you'd see it at Ranger level 4, 8, etc, for the animal companion.