"Let's eat the lunchbox and go outside."
The te is required, because it forms the て form of the word 食べる (to eat). In earlier lessons we learnt individual sentences such as: お弁当を食べます。外に行きましょう。 You can use the てform to connect these sentences together - you swap out "食べます" with the te form of the verb: "食べて" to form the connected sentence above. It's worth having a look online at how to do the te form and when it's used as there's many different use cases!
I'm pleased to hear it! Just to expand on that as reading back I think it needs a bit more info - when we connect the two sentences together using the て form we also take the ending of the second. So in this case the separate sentences translate to: "I will eat lunch. Let's go outside" whereas with the て form it's "let's eat lunch and go outside", I.e. Taking the "let's" and applying it to both.