Data exported to MS Excel contain HTML tags.
MS Excel cannot properly render HTML inside cells. It can render simple formatting like bold text and colors but it cannot render paragraphs, pictures, tables, links and rich formatting in general in a single cell. Wiki markup has the same problem when exported to MS Excel. There's more information about this issue here: https://jira.atlassian.com/browse/JRA-14705.
Export to PDF instead of Excel
You can export to PDF. This will keep all visuals including rich formatting, images, links.
JIRA PDF View plugin supports JEditor-enabled fields:
JEditor also provides a free PDF export option. You can find it on the Issue Search screen (only in List view). This feature is very basic and can only export columns that are visible on the Issue Search screen.
Remove HTML tags after export (using Excel find and replace)
One more workaround is to remove all tags after exporting data to MS Excel. Select all columns that contain HTML tags and then find all tags using the wildcard character - "<*>" and replace them with the empty string.