How auto load items on page Taxonomy page will have sidebar which will have taxonomy menu items and grid layout with image and brief details. 

For example, taxonomy items have brand by certain names, price range, etc. User click on brand name, the grid layout will auto load the brand products without have going to /products/brand_name. On page, it will be /products, and then every brand name will show different products automatically loading on same page.

Is there such a way to do this? Or there is a module which can do that?

Example I found is 
http://www.cynosure.com/products/

Click on any from menu on left column, you will see the products reloaded by menu click.
This can be achieved using exposed filters feature of the Views module. Here is an article explaining this process in detail.

http://www.bryanbraun.com/2013/08/06/drupal-tutorials-exposed-filters-with-views

You would also need the Better Exposed Filters(BEF) module for displaying the filters as links.