Translation:He was touching his new car, when she came out of the house.
Keyword is "sin". It does not exist in English, where it just becomes "his" and thus can refer to "his" (subject's) own car, or some other man's new car.
In Danish, using "sin" always means that the following noun is "subject's thing", that is, owned by the subject. The Danish sentence's use of "sin" makes it certain that the car belongs to the "han" in the beginning of the sentence, because the "han" is the subject.
If it was some other man's car, the Danish sentence would read "Han var ved at røre ved hans nye bil..." or alternatively, if your answer should be correct "Han var ved at røre ved hendes nye bil...".
It's a confusing aspect of the Danish language, and feel free to ask for clarification if my explanation was too unclear.
It is a bit offtopic here, looking at whole "sin" discussion, but "hendes" is used to show possession by a female, e.g. "det er hendes bil" = "it is her car", while "hende" is used in sentences like "jeg så hende" = "I saw her". It is the same as with ham/hans, when we talk about males.
I don't think the word "touching" is what is wanted here. I think maybe 'stroking' is a little more accurate, although that sounds a little awkward in the context of a car.
If he was literally touching the car, it would be more natural to say something like "His hand was on the car" or maybe "He was leaning against the car". Used in this context, "touching" doesn't have the sense of an action, ie. of being actively involved in doing something.
On the other hand, the command "Don't touch the car!" implies an action. But another way to say this would be "Keep your hands off the car!" So touching is really about contact and not about an action.
Consider these sentences:
At the traffic light, two cars were so close to each other they were touching. His comment touched on the subject of fake news.
I assume 'ae' means American English? I don't see how that's relevant.
I agree that 'stroking' isn't great here, but it's hard for me to envision exactly what is meant in the original sentence. It just seems like an odd thing to say.
Another thought: 'Touching' in the progressive is usually not used as an action verb. Sure, you could say "He was touching the car all over", which might mean he was moving his hands all over the car in repeated motions. But, usually the progressive is used for something more passive like this: "They were sitting so close their knees were touching."
If you want to convey a more active sense, I'd say a non-progressive aspect would be better, eg. "He touched the car when she came out of the house." And, if you want to maintain the progressive sense, then you could say "He had his hand on the car when she..." But "he was touching" here is unclear and even confusing. At least in English (whether American or otherwise).
I wish I had a better sense of exactly what this means in Danish because I suspect it's different from what is being translated. Either that or it's just not a very good example for pedagogical purposes.
"Ae" is danish and means to stroke/caress/pet. He caressed the car - han aede bilen. He was caressing the car - han var ved at ae bilen". "Var ved" means he's doing it or "var lige ved" means he almost or was about to. The "lige" conveys the "almost". The rest is context. Touching means he was in the process of touching it.