When people are uncomfortable with your emotional vulnerability it is great to get curious.
What is their relationship with their own emotions?
How were emotions dealt with in their family of origin?
What did they learn in their family about emotions and how they were to be expressed or repressed?
How did they learn to deal with their own emotions as a child and then as an adult?
What might they need to hear or be reassured about when you express emotion?
Let me know in the comments if this is useful.