Adding to xSpuky9's answer, if there's any grammatical confusion about "și": it is not the conjunction "and", but the pronoun "își" (unstressed dative, third person, plural). Next to the auxiliary "au", it must form the mandatory contraction "și-au".
THAT helps enormously also. I really thought it was the conjunction, "și". At some point I am really going to have to sit down and learn gen/dat pronouns and all their glorious contractions. Not even close to confident enough with them yet... Have a lingot also.
It's not obligatory. You could easily say "Ele au inceput ziua razand".
As a native, it feels like putting "si-au" instead of "au" makes the action seem centered on them and how their day in particular was going.
If it was "Ele au inceput ziua razand", it would translate to "They started THE day laughing".
Also, Duolingo likes to put the pronoun when in actual speech it is only used when it's not clear who we're talking about. If the previous sentence is also about them, you would normally say "Si-au inceput ziua razand", without the pronoun.