It's quite funny actually. I was terrible at these puzzles until I started writing this answer. To demonstrate the strategies that I completely and suddenly winged out of nowhere, I will solve the given puzzle.
Firstly, I would like to note the following: Flow puzzles SOMETIMES have multiple solutions, and to my knowledge, these alternate solutions almost always involve a "2x2 pool", like so:
These pools can be used to fill space to get the full score for the puzzle (The entire grid should be filled by the end). The canonical solution to the puzzle will never include these. In fact, in the official rules of the puzzle genre Flow was derived from, these pools aren't even allowed! Much of our strategy will involve the abuse of this rule.
Strategy 1: WHY WOULD YOU EVER DO SOMETHING ELSE???
WHY WOULD YOU EVER DO SOMETHING ELSE??? is the most basic of strategies. Essentially, this strategy applies when no other line can even remotely be justified. We can immediately apply this:
Using WHY WOULD YOU EVER DO SOMETHING ELSE???, I have solved the purple line in the top right of the grid. If you are not convinced, notice how if any other line was drawn to connect them, either a 2x2 pool is created, or the line will close off part of the grid to the rest of the lines. By going out of our way to make this line more complicated, we make the rest of the puzzle harder and nothing else. Therefore, WHY WOULD YOU EVER DO SOMETHING ELSE????
Strategy 2: Who owns me?
In other words, look at an empty cell in the grid. Which line "owns it"? Which line can be the only one to pass through it?
I have already applied WWYEDSE??? to solve the blue line (Elaboration: You can consider the case in which this isn't the solution and notice a contradiction concerning closed loops). More importantly, notice the empty cell marked by the red X. It must be "owned" by the dark red line! This solves dark red. To further see this, consider the case in which this cell is NOT owned by dark red. Then, this cell cannot be owned by anyone! But we KNOW that in the intended solution, it must be owned by someone! Thus a contradiction.
Literally the exact same logic applies to the cell marked by the green X. This cell must be owned by dark blue. We repeat this logic to get a partial solution to dark blue:
I have applied WWYEDSE??? to solve the dusty color.
You may be saying, "GREENTURD! THE REST OF THE DARK BLUE LINE IS OBVIOUS! JUST DO IT ALREADY YOU FLIPPIN HALF COOKED PANCAKE!!!1!!1" Now, I'm sure with enough thinking and considering ALL possible cases, that you could apply WWYEDSE??? to solve dark blue... but it's hard to be certain. Instead, I would like to demonstrate...
Strategy 3: HYPOTHETICAL CORNER LINE PARTY!!!
Imagine that you start a new Flow puzzle. Now, imagine a corner of this puzzle... zoom in. See that corner cell? How it would be wonderful to know who owns it. But sometimes, it is unclear.
So, let's invent a new color that can be whatever the hell it wants to be. Let's call it Color X. We define it as the color of the line that passes through this corner cell.
How does this help? Think! If Line X owns this cell, what other cells must it own? Yes! The two adjacent edge cells! And wait: WE CAN APPLY THIS EVERYWHERE! Hence, we have created a HYPOTHETICAL CORNER LINE PARTY!!!
"But greenturtlez, that doesn't look like much of a party... the lines who were invited don't look like they are having fun! :("
It's just getting started! We can abuse the "no 2x2 pools" assertion to extend these lines, alot!
Now halt! Do... you notice anything? Look at the "orange" line at the top right. It can't connect to that weirdo pale color under't... so it must connect to... OMG "BROWN" AND "ORANGE" ARE PART OF THE SAME LINE!!!
And look! Their connection created a corner! Which set up another HYPOTHETICAL CORNER LINE PARTY!!! Corner lines create MORE CORNER LINES! Now, we use WWYEDSE??? and Who owns me? to deduce even more:
The cards are set, our hypotheticals and certainties in place... and then: BOOM!!!
Hm, that pink line... I do wonder. It has to own that "brown" line... but to what end? Wait a sec... is that a light I see? Oh lord. Everything that seemed strange, all that extra space we deemed so awkward, it all adds up to the one conclusion:
And with that, the puzzle is solved:
2x2 pools are not intended, so pretend they are illegal, and use this to your advantage.
If you see a possible way to solve a color, and all other ways to connect the dots have absolutely no justification whatsoever, just do it.
When you are lost, find a big mess in your puzzle, and look at an empty square... Think about which line MUST pass through this square. If you can point your finger at only one color that can "own" this square, you have made a valuable deduction.
Finally, when all is lost, draw hypothetical lines, that you know must exist as part of a line, but you do not know which.
Have fun solving these puzzles!