"She looks like she needs to wash herself."
Translation:Hon ser ut att behöva tvätta sig.
Late answer, but I'll give it a go since no one else has.
I don't think att is a conjunction in the main sentence here (Hon ser ut att behöva tvätta sig), I'd say it's just the infinitive marker. So what happens in this sentence is that a verb in the present (ser ut) takes a verb in the infinitive, like in Hon tycker om att äta 'She likes to eat'. Verb + verb in the infinitive. This could also happen without the att, like Hon verkar vara trött 'She seems to be tired'.
In the alternative translation Hon ser ut som om hon behöver tvätta sig (Hon ser ut som att hon behöver tvätta sig is also possible) we have the conjunction som om, which joins the clause and subclause (the subclause is hon behöver tvätta sig), then we get two distinct clauses that both have finite verbs.
Hope this helps. Maybe what's tricky here from an English speaking point of view is that it could be a bit unexpected that a verb like ser ut som can take an infinitive, when looks like cannot.
You mean something like Hon ser ut som hon behöver tvätta sig? I believe that works, and if it doesn't, I would really like to know why not.
I think your question about why Hon ser ut som att behöva tvätta sig requires the att is a little clearer if you translate ser ut som as "appears" instead of "looks like". Hon ser ut som behöva tvätta sig would be "She appears need wash herself". Hon ser ut som att behöva tvätta sig would be "She appears to need wash herself".
Note that a lot of English speakers would naturally say "She appears to need to wash herself", but the second "to" is not needed for correct grammar.
Maybe tendency isn't the best of words. What I mean is that "verkar som" means "seems like".
If the following word is a verb, you don't need the att. You can say "hon verkar vara rysk". But if you introduce a subclause, you need to say "det verkar som att hon behöver tvätta sig".