Directions on a treasure map
A treasure map says '3 steps east, 2 steps north.' Each direction is a vector, and combining them — some east, some north — reaches any spot on the map. A linear combination is just mixing basic direction-steps to get anywhere.
Vectors add componentwise
A vector is an ordered list of numbers. You add vectors component by component and scale them by multiplying every component by a scalar.
Span
The span of a set of vectors is every vector you can reach using linear combinations c₁v₁ + c₂v₂ + …
[1,2] + [3,0] = [4,2] 2·[1,2] = [2,4]