These pronouns are not there to clarify the meaning but to set it. Removing them is possible but significantly affects it:

Qu’est-ce qui s’est passé ? : What happened? (some event)

Qu’est-ce qui est passé ? : What has moved/gone? (some object)

Je me suis lavé. : I got washed (literally: "I washed myself").

Je suis lavé. : I’m washed (I’m clean)

Nous nous sommes habillés. : We got dressed (some time ago).

Nous sommes habillés. : We are dressed (now) (we are not in underwear/naked).

Note that je suis lavé and nous sommes habillés are ambiguous, as lkl answer shows. I picked the most likely meaning where lavé and habillés are adjectives attributs du sujet. These sentences migth also be understood to be passive voice verbs:

Je suis lavé. : I’m getting washed (by someone else)

Je suis habillé. : I’m being dressed.

The real subject is often explicit in passive mode: je suis lavé par l’infirmière, je suis habillé par un styliste.

As Jlliagre explained, the meaning does change without the reflexive pronoun – and the grammatical construction often does too.

Qu’est-ce qui s’est passé / Je me suis lavé / Nous nous sommes habillés = passé composé of pronominal verbs

Qu’est-ce qui est passé is still the passé composé, albeit of the non-pronominal verb passer

However, Je suis lavé / Nous sommes habillés is a different grammatical construction altogether – it’s the passive voice in the present tense.