My first thought is that using the singular "robot" is incorrect. You should either say:
I gradually became fascinated by robots because I started to observe the close connection between them and the developing technology.
I gradually became fascinated by robotics because I started to observe the close connection between it and the developing technology.
These sentence are fine. There are no more changes to it that need to be made. There are some changes you could make. For example, using the word "observe" is perfectly fine here. You could use an alternative, like
...because I started to notice the close connection...
...because I started to realize there was a close a connection...
But "observe" is grammatically correct, and it does not sound arrogant in any way.
Using "became" is correct for this sentence. If you had said "become" it would sound like you are talking about how you are at this very moment becoming more fascinated by robotics. Also other parts of the sentence would need to change. For example
I gradually become fascinated by robotics as I start to observe the close connection between it and the developing technology
And you could use "developed", but again you would need to change a little bit of the sentence. It would not be correct to say
I gradually developed fascinated...
but you could say
I gradually developed a fascination for robotics...