There are a few ways that we know. First is just the conversation surrounding him. As you said, many people in the historical record alluded to his sexuality. We can guess that there was quite a bit of gossip surrounding that. But there are other, more concrete historical sources that we can use.

Aside from the hearsay, we know from Röhm's autobiography, Die Geschichte eines Hochverräters (The History of a High Traitor), that his views of morality differed quite a bit from that of the official morality of the National Socialists, especially regarding his views of homosexuality. He had interesting views of masculinity and homosexuality and lamented the suppression of homosexual desires by the National Socialists.

This, on its own, does not necessarily make him a homosexual, but we know from some of his letters (especially those to Dr. Heimsoth) that he only wrote it in this vague, separated voice at the urging of his friends. In these letters, he admits to being a homosexual. Furthermore, he confirms the suspicion that the advocacy for homosexuals in his autobiography was, in fact, a close personal subject.

So, we have a few things here. One is the confirmation for historians that yes, he was a homosexual. Second, we have the reasoning behind potential assumptions by his contemporaries. Lastly, we have confirmation that these assumptions were made by his contemporaries and that it was spread around like gossip.

Source:
Hancock, Eleanor. "'Only the Real, the True, the Masculine Held Its Value': Ernst Röhm, Masculinity, and Male Homosexuality." In Journal of the History of Sexuality 8, No. 4 (April, 1998): 616-641.