Motivating the Implication Operator
We do not take implication as some form of causality, but as a form of logical consequence, i.e., that the truth of a first thing forces us to accept that a second thing is true. The essence of "and" is: that one thing and another are true necessitates that each of these two things is true. So with $C\land D$ representing the statement that $C$ and $D$ are true forces us to accept that $C$ is true (it also forces us to accept that $D$ is true). If you do not agree, you may have an unusual notion of "and" (or of implication). And this fact, i.e., that the truth of $C\land D$ forces us to accept that $C$ is true. The accepting-force behind this argument is always correct, no matter whether $C\land D$ itself is actually true. This now precisely what $(C\land D)\Rightarrow C$ symbolizes.
A more standard truth table:
I don't see what $C\land D \implies C$ has to do with it, but the truth table for IMPLIES can be justified as follows using a form of natural deduction:
