Gravitational waves are fluctuations in spacetime, but what is spacetime? Your intuitive sense of space and time tells you what space and time ARE. In science, and physics, in particular, we use words in very specific ways. In physics, spacetime is a combination of distance and time. So, when we say that gravitational waves are fluctuations in spacetime, we are saying fluctuations in distances and times.
Gravitational waves are a prediction of general relativity (GR), a theory of space and time (spacetime), and matter and energy. We talk about observers in GR, and ask questions like, "What would an observer observe when a gravitational wave passes?" Now, you say, "The observer observes fluctuations in spacetime, distance and time, of course." And what does that mean?
It means that the distances between points increase and then decrease and then increase and then decrease, etc. The increases and decreases are above and below, respectively, the distances before and after the wave passes. The same is true for the time between events, like the time between ticks of a clock. As the gravitational passes the observer, the observer's clock runs faster than normal, then slower than normal, then faster than normal, etc.
And what is it spacetime instead of space and time? That's because in GR, and in special relativity, which is a special case of GR, spacetime are unified. What does that mean? It means that observers at different positions and with different velocities relative to each other measure distances and times. One observer may see two events happening simulataneously at two different locations, while another observer sees the two events happening at the same location at different times. So space and time flow back and forth depending on your velocity.