The most common and most likely answer to this crossword clue is the 4 plus 2 letter word DIAL UP
I've seen this crossword clue in The Daily Telegraph
D
I
A
L
U
P
Buddy explains!

The straight part
We know that "Make connection with" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make connection with = dial up.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DIAL UP mean?
DIAL UP noun- A network connection that requires a telephone number to be dialed.
- (of a connection to a network, especially to the Internet) that requires a telephone number to be dialed