Learn Portuguese (Portugal): A gente estende a manta na relva. - We informal lay out the blanket on the grass.

This is the core feature of a gente. Even though it means 'we' in informal Portuguese, grammatically it acts like a singular noun (literally 'the people'). Because of this, it always takes the 3rd-person singular verb form: estende instead of the plural estendemos.

Is it also correct to say nós estendemos a manta na relva?

Yes, absolutely. Nós estendemos is the standard way to say 'we lay out' and is perfectly correct. Native speakers just use a gente very frequently in casual, everyday conversation, so it is important to be comfortable with both.

I learned that estender means 'to extend'. Why is it translated as 'lay out' here?

While estender literally means 'to stretch' or 'to extend', it is the standard Portuguese verb for spreading something out flat. You will commonly hear it used for laying out a blanket (manta), a beach towel, or even hanging out the laundry (estender a roupa).

Why use the contraction na instead of just the preposition em for 'on the grass'?

In Portuguese, you almost always need to include the definite article when referring to a specific physical location. You combine em (on) with a (the feminine article for relva) to create the mandatory contraction na.