Bookeo lets you collect reviews from your customers, and these reviews can be published on your web site. Click here for more information.
TripAdvisor and others do not let businesses post reviews left by their customers, for security reasons. It would be too easy for a business to post fake reviews. So the reviews that customers leave on Bookeo cannot be 'pushed' into TripAdvisor and similar sites.
However, you can ask customers to write a review on Tripadvisor, Yelp, Google Places, etc.
1. Get the badge/link to write a review for your business from TripAdvisor, as explained in their help guide.
2. Copy and paste that code in your Thank you email, in Settings>Thank you email.