We learn on Pottermore that muggle-borns are found by a magical quill in Hogwarts. They then receive their letter, which identifies them as a wizard, and they join the magical community. What did they do before Hogwarts? How would they have found out that they were a wizard?
Let's not forget that HP takes into account mythology and history, like Merlin who attended Hogwarts back in the late 900s, so we know Hogwarts is suuuper old. And before that there were druids, witches and magical creatures who lived among the Muggles, even helped them or were persecuted by the same.
Looking historically, Islam forbade the use of magic and it came in the 6th/7th century, so we know there were magical folks before. Looking back to the ancients and early civilizations - these and their religions were formed around people with magical abilities to see into the future or communicate with Gods, make wonders and deal with supernatural creatures.
Sooo, to get back to your question (because I could write a book on magic in real history): people with magic, muggleborn and purebloods and all the in-between, even before the foundation of Hogwarts, were able to grow and use their skills.
I can't find anything in canon about this, so this will all be guesswork.
Hogwarts was founded in the 10th century, so it's been a while since then. I haven't seen anything about Muggle-borns from back then, so it could be the case that the appearance of magic in Muggles' children is a recent occurrence and back then it was only purebloods and occasionally halfbloods. If, though, there were Muggle-borns at this time, remember three things:
1.Without training, wizards and witches are never able to do any magic except accidents, like vanishing the glass at the zoo, or jumping on to the roof of the kitchen. Possibly, Muggle-borns would just never be seen for what they were and would go through their life untrained, believing that they are a Muggle.
2.This was before the Secrecy Laws. Muggles might have known about magi and magical people at this time; maybe they just sent their kids to train with another wizard.
3.Remember what I said in the first one: without training, wizards and witches are never able to do any magic except accidents. It seems highly unlikely that there was no place for wizards and witches to go--some school that was a predecessor of Hogwarts. If so, Muggle-borns could go there. If not,wizards and witches probably trained in individual apprenticeships.
There was also Queen Maeve, a possibly muggle-born witch before Hogwarts who trained many wizards and witches, including, I assume, muggle-borns, in Ireland.