They are the rulers of the Reach:

The Tyrells control the most fertile agricultural land in the entire realm. Most of the food and civil supplies of Westeros come from land under the supervision of House Tyrell. Lady Olenna even threatened Cersei Lannister that she will starve out the crown if something happens to her grand-daughter and grandson.

They have a massive Battle-tested army at their disposal:

Tyrells command the largest army in Westeros. In addition, they have a huge fleet of ships from the Redwyne, which is as big as the crown’s fleet. If they really want to dethrone Lannisters, they can get it done in a week or two. They are legitimately powerful.

Lannisters are dependent on Tyrells to retain control at King’s Landing:

At present, there is tension simmering between the Lannisters and the Tyrells, after the arrests of Margery and Loras. The alliance is fragile. Lannisters’ control over King’s Landing is weakened and Tyrells are upset. In the Battle of the Five Kings, it’s the Tyrells which saved the crown from Stannis’s forces invasion.

Daenerys Targaryen needs Tyrells on her side to even dream of sitting on Iron Throne:

Dany is caught up in the east at the moment, in the last scene, she was surrounded by a horde of Dothraki warriors. Once she unites the east, she can begin to march towards Westeros, with a significant army rallying behind her. But if she really wants to sit on the Iron Throne, she needs the Tyrells on her side. As Tyrion told her in season 5, at the moment only Tyrells can be convinced to support her claim.

House Tyrell is one of the richest and wealthiest in Westeros:

They didn't start out as being powerful or wealthy, but now, they are the richest, in terms of assets or monetary wealth, as Lannisters gold mines have begun to dry up.