It's a slang. "떳다" means "appear".

The root of 떴다 is 뜨다, which is intransitive verb and originally means to float physically on the suface of fluid or to maintain (itself) in the middle of liquid or gaseous material(s) without sinking. For example: 1. 일반적으로 금속은 물에 뜨지 않는다 : In general, metalic materials don't float on water. In 어머니 떴다. 뜨다 is used to describe a situation that someone/something appears abrruptly without giving prior notification. However, it means implicitly that the speaker has been waiting for the emergence of someone/something to give a warning to others who are in the same place and who belong to the same group of the speaker. In some case where the speaker is alone, he(she) might be warning to himself(herself). However this usage of 뜨다 is quite informal and inappropriate for writing. The other typical usage of 뜨다 is to describe someone/something that gets popularity. For example: 이 (남자) 배우는 최근에 뜨고 있다 : This actor is getting popularity recently. However this usage is also quite informal, so only used in spoken Korean among close people or TV shows.

Original form is '뜨다' . '떴다' is the past tense. When the meaning of '떴다' is 'appear' or 'come out' , it's a kid of slang but popular.

korean dictionary is not very helpful because verbs can have many different forms depending on the endlings. When you look up something in a korean dictionary you have to type the exact base form. Also korean dictioanries don't seem to keep up with korean people. 뜨다 is the base form, and you have to remove 다 and add an ending. Here the ending is 었다. So 뜨다 + 었다 = 떴다. Whenever you see ㅆ as a final sound, it means it's past tense. 뜨다 has many meanings but usually refers to "to emerge above or to rise" . 어머니 떴다 means my mother emerged(appeared). 떴다 in this usage is slang. Young people often say it to friends.

Yes, I'd like to give one more example. “짭새 떴다”, 짭새 is also a slang which means the police. Imagine. There are some teenagers drinking alcohol and suddenly several policemen coming near them. Luckily, one boy found them and whispered “짭새 떴다! 짭새 떴다!” What did he say it for? Because he wanted to hint his friends will be under police observation soon. His friends will know “Hide alcohol!” was what he really wanted to say even though he didn't say it. A basic form of “떴다” is 뜨다(has many meanings, one is ‘rise’). I guess you will be able to understand it if you think the sun lights up everything with rising. Hope this helps.