Here's an illustrated version of Sharkos' answer. We'll use the disk model of ℝP2: consider a disk with antipodal points on its boundary identified.
We can imagine a closed path that starts at the center, goes straight up (1st part in light blue) hits the topmost point of the disk's boundary - which is also the lowermost point of the disk's boundary - and then continues upwards (now from the bottom, 2nd part in dark blue) to reach the center again and close the loop:
We cannot shrink this loop to a point: we can modify it within the interior of the disk, but that doesn't get us very far. And if we try to change the loop at the point it crosses the boundary, we have to move the top and the bottom of the loop so that they cross at antipodal points, otherwise we brake the loop. For instance, doing the following breaks the loop:
We can slide both points so they stay opposite each other:
But that doesn't allow us to shrink this loop to a point.
Next we consider a loop that crosses the boundary twice:
The loops starts from the center, goes straight up to the top of the disk (1st part in light blue), comes up from the bottom to the center (2nd part in dark blue), continues up to the top of the disk (3rd part in pink) and finally comes up from the bottom of the disk back to the center to close the loop (4th part in red).
As in the previous example, if we move a point where the path crosses the boundary we have to move the opposite point in the opposite direction so they remain antipodal to one another. We can start to rotate the 1st and 2nd parts of the path counter-clockwise in a continuous way until it looks like this:
The loop is not broken in any way. We continue to rotate it:
Until we've rotated the first half of the path 180 degrees:
And this is where the magic happens: if we imagine a particle following along the loop, it travels down part 2 (in dark blue) and when it reaches the center it turns around and travels up part three (in pink). That means that we can pull the point where the end of part 2 and the beginning of part 3 meet upwards without breaking the loop:
We can now pull the 2-3 part of the loop past the boundary:
At which points it's evident that the loop is contractible.
As Sharkos said, it's sliding points around on the boundary that messes with our intuition: when there is only one point where the path crosses the boundary we cannot shrink the loop to a point because any time we move that point it must be "reflected" in the opposite side of the circle (this is more of an appeal to our intuition rather than a rigorous proof). When the path crosses the boundary twice we can slide one pair of opposite points around to effectively "untangle" the loop.