If while trying to browse through the App store on your iPhone, you continuously receive the error 'Can't connect to app store', you can find the resolution for the error below.
The following screenshots have been taken on an iPhone 6 Plus running iOS 8.3.
Go to iPhone Settings app.
Settings are displayed. Scroll down till you see iTunes & App Store
Click iTunes & App Store
iTunes & App Store settings are displayed. Click on your Apple ID at the top. (In my case its email@example.com)
A small menu is displayed. Click Sign Out.
You are signed out of iTunes & App Store. Enter your password again in the password field.
After you enter your password, the Sign In button becomes enabled. Click Sign In.
You are again signed into iTunes & App Store. Go back to the App Store. You will find that the 'Can't connect to app store' does not appear again and you are able to browse through apps again.
If the above resolution did not work for you, you can try changing the DNS for your Wi-Fi connection to OpenDNS or Google DNS using the steps given in the following link:
Do let us know in the comments section if this is able to resolve the issue.