What makes it hard is what it forbids
The rule is easy to state and surprisingly restrictive. Every cell must be used, so the line cannot afford to leave anything behind. It cannot cross itself. And it has to arrive at 2 after 1, at 3 after 2, and so on, which pins down the order of the whole journey rather than just its endpoints.
Together those mean most directions you could go are wrong, and the puzzle is the business of working out which. You are rarely asked where the line goes. You are asked where it cannot.
The three things to look for
Start with forced steps. A cell the line has entered, with only one unused neighbour, has only one exit. Chains of these run for a dozen cells sometimes, and clearing them first shrinks the real problem enormously.
Then watch for stranding. Because the line must reach every cell, any move that leaves a cell with no possible way in or out is wrong - and a corner or an edge cell will strand itself long before the middle of the board does. Scanning the edges is usually more productive than scanning the centre.
Last is splitting. The remainder of the line is one unbroken walk, so it cannot jump a gap. A move that cuts the untouched cells into two separate pockets can be ruled out immediately, however reasonable it looks locally.
A note on the family it belongs to
Mathematically this is a Hamiltonian path - a route visiting every vertex of a graph exactly once - with the waypoints acting as ordering constraints. That is genuinely what it is, though nobody thinks about it that way while playing, and the general problem being famously hard is precisely why a well-constructed board is satisfying: the constraints are what make it tractable by hand.
Puzzles of this shape turn up under a variety of names in newspapers and apps, usually with numbers scattered through the grid and a line to be drawn between them. The core is always the same: one path, every cell, numbers in order.
Every board here is solvable by reasoning alone
This genre is the easiest of the three to construct badly, because it is trivial to produce a grid whose only route to the answer is exhaustive search. Those are unpleasant to play and they are not published here.
Each board is verified to have exactly one path, and separately checked by a solver restricted to the named deductions above. Anything it cannot finish without searching is discarded. The tier is the hardest deduction the board actually requires.
Common questions
Does the line have to go through every cell?
Yes, every one, exactly once, and it may not cross itself. That is what makes the puzzle tight - the line cannot afford to leave anything behind.
What are the numbers for?
They are waypoints, and the line must reach them in ascending order. They fix the order of the whole journey rather than only where it starts and ends.
Why can I not finish a board that looks nearly done?
Almost always a stranded cell - somewhere the line can no longer reach without breaking a rule. Check the corners and edges first, because they run out of options long before the middle does.