Insecure people react badly to change in relationships.
For example, I have one friend that has been my friend for years, 7 years. In all that time, I have always known that she was a reactional friend. I could always count on her for reactions. I mean, jubilant, bubbly, over-the-top O.M.G. reactions. As we have grown and she doesn't necessarily react over the top to everything, our relationship has changed. She challenges me more to account for my reactions or why I am putting up with others and at first, it was difficult to deal with. At first, I got agitated by the fact that I was getting the reaction that I used to. This was as a result of my being insecure over our friendship when she started treating me differently.
As Joyce points out, we can seem blessed in relationships when we are getting what we want, but when things change, we realize we weren't secure but spoiled. I did great with my friend as long as I knew what to expect from her. I knew that she was not someone who I could go to cry on her shoulder. She was someone who I would call as soon as I had good news or needed an over-the-top reaction to feel validated. But, when that changed, I found myself avoiding her. I no longer felt validated, I felt criticized. (Luckily, both of us are too stubborn and we are still friends to this day.)
Feelings of insecurity in relationships mainly results from being unsure of one's self, looks or anything else you consider important. If you don't feel that you are good enough for your friends or partners you always think that he/she will be looking for someone to replace you.
If you are scared that he/she is looking for someone to replace you, then you become jealous, needy, and clingy and most relationships can't handle that. Try to relax and not put so much pressure on your relationships.
**by the way, I used a friendship as an example here to make a point: Insecurity doesn't only effect romantic relationships.