I actually met and went hunting with a great-great grandson of Emperor Frederick II this week, and was very excited, so forgive my late answer of this question.

I also have on hand Frederick's De Art Venandi Cum Avibus, which gives ample evidence that this non-answer was probably a high compliment.

Batu Khan is a grandson of Ghengis Khan, who was probably-but-not-assuredly a falconer. Another grandson of Ghengis Khan's, Kublai Khan, was definitely a falconer, though, and lived in a culture of high falconry. Marco Polo writes, speaking of Kublai Khan,

He takes with him full 10,000 falconers, and some 500 gerfalcons besides peregrines, sakers, and other hawks in great numbers; and goshawks also to fly at the water-fowl.

From The Travels of Marco Polo, Book 2, Chapter 20

Batu Khan was born only 8 years before Kublai- it isn't as though Kublai simply invented falconry from thin air, and it's likely that Batu was also at the very least informed on falconry and bird-hunting. Batu founded the Golden Horde, which also had a later reputation for high trade in, and use of, falconry birds, which was very likely present since inception and carried the same high status.

Now, this brings us to Frederick II: an emperor both famous and infamous for his love of falconry.

You have to understand that Frederick was fucking obsessed with falconry. He was frequently mentioned as "sneaking off" to go hunting, sometimes ignoring official duties or taking impractical courses of action just to allow himself more time with his hawks or to visit certain hunting lodges or castles.

He would often judge others on their ability to tame, handle, or hunt with hawks and goes on at length in his book, essentially to express admiration for someone who was a good falconer or distaste for someone who was a poor one. He might have even considered it a character trait.

When Frederick II responds that he thought Batu Khan would make a good falconer, for he understood birds, I think- knowing what I do of Frederick's personality, and judgmental attitude both positive and negative when it came to a person "knowing birds"- he meant that very plainly and literally as a compliment. Around the same time, Frederick's public statements were (of course) decidedly against the 'savage' Mongols, but he expressed great admiration for their military prowess and was hesitant to aid those the Golden Horde were antagonizing (because of layers of Hungarian and Papal politics that are well beyond me.)

In short, I think Frederick was more interested in going hunting and talking birds with Batu Khan than he was in outright war against the Golden Horde.

For what it's worth- the letter also went unanswered, and Batu likely never got the compliment, but what a world it would be if Freddy and Batu had done a month of amicable hunting together?