A bit late to the party, but I'll try anyway!

So, first, speed is distance over time. Miles per second, kilometers per hour, whatever.

Consider a person rocketing by a planet in a little spaceship at a good fraction of the speed of light. To amuse themselves, they're bouncing a ball between two paddles on opposite walls of their craft. The ball describes a path like:

O--------O

--O----O

-----O

Of course, to a person on a planet they're blasting past, the path looks different - the ship moves a long way between each bounce, so they see:

O----------------------------------O

-------O------------------O

----------------O

The thing is, both of these are correct from each point of view - from each reference frame. For the shipboard person, the ball moves the width of the ship, and for the planetside person, it covers the distance the ship traveled in the bounce (plus some for the width).

Now, swap the ball for a photon, which always moves at the same speed. The distance the photon travels from the two points of view - the two reference frames - is different, so the time component of the photon's measured speed must change as well because the photon's speed remains the same! Each side sees the photon moving at the same speed, despite the difference in distance traversed each pov sees - which means each must also have a different measurement of the time involved!

So, time is compressed on the spaceship relative to the planet - from the ship, the planetside observer is moving very fast, while to the planetside observer, the space pilot is moving in slow motion. The speed of the photon is universal - it's the distance it travels between bounces, and therefore how long it takes to bounce, that differs between their perspectives.