Things are simple in theory: in order to level up any unit, you will need to have two other units of the same star rating. So in order to level up a unit to two stars, you will need three 1-star card of the same type. In order to get a unit to three stars, you will need three 2-star cards of the same type. In other words, getting any unit to 3 stars requires a whooping 9 cards of the same type. The leveling up process happens automatically, as long as you meet the requirements.

Usually, at two stars, a hero deals double damage and has twice as much health, while at three stars the base stats are 4 times greater. So we can say that yes, you get great benefits from working on starring up your units.

However, the problem is that getting a hero unit to three stars is very difficult and expensive. The higher the tier of the card, the more difficult it will be: for a 4 gold card, for example, you will need a total of 36 gold to get to three stars, plus something close to that amount for refreshing the cards you got in order to actually be able to purchase the said cards.

Based on my own experience, I would say that the easiest cards to upgrade - while still being good in the later rounds - are the Tier 2 and Tier 3 ones (2 and 3 cost heroes).

Tier 4 and above are really difficult to get to more than 2 stars, while Tier 1 cards at 3 stars are obviously weaker than T2 and T3 units at the same star level.

So as a strategy when it comes to leveling up your heroes in Dota Underlords, I would say that you should try and get your main Tier 1 unit to three stars, then 2-3 T2 or T3 heroes to three stars. Get Lycan and Treant Protector to three stars, for example, and you're doing great!