I am totally new to GIS. I have a QGIS project. I am adding the screenshot.
As you can see the map is sort of rectangle in projection. How can I project it like this (without the grids)
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The shapefile from Natural Earth contains the south pole. The Mercator projection is not able to render that point for mathematical reasons (it would be in inifinity).
What you can do:
If you want an exact square, cut off the antarctic until 85.01 degrees south.
Looks like you want a Mercator Projection. Try to save-as your shapefile assigning the WGS 84 / Pseudo Mercator (EPSG: 3857) projection or assign this projection to your project (using transformation on the fly).