What are the bug tracking tools helping to take screenshots of bugs occurred?

My response is heavily biased towards web applications, and in-browser bug reporting, though some tools may allow native iOS and Android implementations, as well.

The answer depends on on the type of audience that needs to report bugs: private and targeted (developers, employees, etc.) or public and open (all website visitors).

Private group typically have some level of vested interest in reporting bugs, and may have higher tolerance for user experience (UX) issues associated with the bug capturing process; public groups, however, need much more streamlined tools and less hurdles in order to encourage bug reporting.

FOR PRIVATE GROUPS, the top two tools that we have used, may be a bit too complex for public groups, though they can integrate with more user-friendly tools. Both of them also allow annotations on the screenshots that they take.

https://getmarker.io/pricing
MARKER starting at about $25 per month is the cheaper of the two. It integrates with some tools such as Trello, Jira, GitHub and Slack, and they have plans for integration with HipChat, which may make it more user-friendly for the public group.

https://usersnap.com/pricing
USERSNAP starting at about $80 per month is more expensive, but has more features and certainly more integrations out of the box.

FOR USE WITH PUBLIC GROUPS, I recommend using a chat service, with built-in screenshot capturing, since those generally seem more user-friendly, and have less complexity in the capturing phase, and have an established use pattern. Of course, some of these make great tools even for more private audience groups, as they can be limited to such groups easily using some basic server-side coding.

https://doorbell.io/pricing
DOORBELL.IO that has a free plan, supports screenshots. and is one that we have not used, but seems simple, very promising, quite open. It is also highly malleable due to having an API for integrating it into your own widgets, and supporting "web hooks" which may be used to post content from it to any web address.

https://www.olark.com/pricing
OLARK that also has a free plan, is one that we have used extensively, since it has a co-browsing feature. Once initiated by the chat operator and accepted by the visitor, it provides a live feed into the visitor's browser.

NOTE: We have been an Olark partner for a while now, though I am writing without bias, and have NOT included our referral link in this response.