She took an educated guess.
Poppy is characterized as someone who is very, very quick to pick up on things and just roll with them, for instance here:
But Poppy, just reaching the top, was totally undaunted. Without breaking stride - in fact, she picked up speed - she ran straight at the portal, planted her hands on the windowsill, did a handstand, let gravity flip as she broke the plane and landed on her feet on the grass, upside-down with respect to Janet and facing her. (...) "Come on!" Poppy said brightly.
(The Magicians Land, p.306)
To reiterate: She ran up a staircase, entered the room, saw a portal leading to a different world, made sense of it, even noticed the fact that gravity was behaving differently behind that portal, while even picking up speed and gracefully jumping through it in a way that even makes her landing on her feet. Janet on the other hand didn't manage to get gracefully through it even though she had much more time to prepare and saw Umber going through the portal - thus being prepared for the gravity change.
Now fast-forward to the discussion with Umber. Poppy had several clues to go on: 1) The telegram told her that
Umber was evil
2) From what she knew, The Beast was probably the evilest thing that ever walked Fillory, considering
it even made its way briefly into Brakebills
3) From the pursuit of Umber and the discussion after, it became apparent that
Umber was not only evil, but completely, truly, Heath-Ledger-Joker-Crazy (He kind of reminds me of Carl from Llamas with Hats)
4) Umber directly mentions a connection
between his faked death and Martin/The Beast:
"We thought You were dead." "Oh, I know!" (...) "But I was just pretending. Martin wanted it that way. Such a strange boy. Never came out of his Oedipal phase, I don't think. He was always talking about his mummy in his sleep, wondering if his father was alive, that sort of thing."
(The Magicians Land, p.310)
5) She also definitely knew that
everyone thinks Martin killed Umber.
because they discuss this right after they discuss the telegram.
To me, same as with the portal, Poppy took all this in stride:
A boy who turned into a beast, an evil god who faked his death after said boy requested it, making it look lile that said boy killed him
which was enough for her to base a guess on.