The thing that made me grok lambda calculus was someone[0] in a tutorial online treated as a simple substitution game. They said: Imagine "λx.chocolate covered x" is a lambda expression, and if you apply that to "peanuts" you get "chocolate covered peanuts" and if you apply it to "ants" you get "chocolate covered ants". I knew at that time that LAMBDA was an operator for creating "anonymous functions" in Lisp you could pass to MAP or APPLY, but the "chocolate covered" thing was what made the penny drop that in pure LC, a lambda is just a template with a variable, and application (beta reduction) is substituting values for variables in the templates.
[0] It was probably Jim Larson: https://www.cs.unc.edu/~stotts/COMP204/Lambda/scheme.html