Imagine a table, aligned to north-south-east-west just for convenience. You have a velocity of 5m/s north-east (specifically at a bearing of 38.67 degrees, which is the correct angle for the 3-4-5 triangle you're choosing to use as an example).
Now look at it from the south edge of the table, so that all you can see is east and west (technically also up-and-down, but we're not paying attention to up and down since everything is on the table). If you looked at it from this perspective, you would see the object traveling at 3m/s east. You couldn't see the north-south portion of its movement because you're looking from the south edge of the table.
Likewise, if you viewed it from the east or west edge of the table, so that all you can see is the north and south movements, then you would see an object moving 4m/s north.
Now the reason we find this sort of splitting-up of vectors useful is that many of our equations for motion are linear functions. This means that we can look at the north-south axis on their own, and look at the east-west axis on its own, and then combine the results to see the entire path the particle takes. Your intuition may be having trouble with this because not all things you could want to look at have this nice linear function property to them. If you're worried about something acting non-linear, then you're going to raise questions about why this splitting up was valid at all. But rest assured, physicists have spent a great deal of time trying to make as many of their equations linear as possible, because it lets them do these sorts of clever vector tricks which make the math much easier!