"(봄이) 왔네요" => (Spring) came / Spring has come / Spring is here.
It is a past tense form of "오다" with the "네" and "요" suffixes.
- 오다: root form for "come".
- 왔다: past tense of 오다, in the "formal-plain(not polite)" ending.
- ~네: sentence ending suffix expressing a light surprise.
- ~요: common ending expressing politeness.
- 왔네: "it came[has come]" with a light exclamation of a surprise.
- 왔네요: 왔네 with a more polite tone.
"약혼자 왔네요" (약혼자가 왔네요 is better) means "My fiance has come[is here]".
It is correct, but may be a little funny sounding.
To understand 왔네요 fully, you need to know sentence endings and verb conjugations.
Korean has highly developed ending forms to express diverse connotations and nuances.
For example 왔습니다/왔다/왔어/왔어요 are the four most common declarative endings.
But they are flat statements which would sound stiff and boring in poetic expressions.
So one might use one of the more subtle endings: ~지, ~네, ~ㄴ데 among others.
If you include those for questions, negatives, etc. there are well over a dozen endings.
So Korean is very expressive in this area (and thus takes longer to master too).