Install Visual Studio 2013 on Windows 7

The minimum requirements are based on the Express edition you're attempting to install:

Express for Web (Web sites and HTML5 applications) - Windows 7 SP1 (With IE 10)
Express for Windows (Windows 8 Apps) - Windows 8.1
Express for Windows Desktop (Windows Programs) - Windows 7 SP1 (With IE 10)
Express for Windows Phone (Windows Phone Apps) - Windows 8

It sounds like you're trying to install the "Express 2013 for Windows" edition, which is for developing Windows 8 "Modern UI" apps, or the Windows Phone edition.

The similarly named version that is compatible with Windows 7 SP1 is "Express 2013 for Windows Desktop"

Source

Visual Studio Express for Windows needs Windows 8.1. Having a look at the requirements page you might want to try the Web or Windows Desktop version which are able to run under Windows 7.

Visual Studio 2013 System Requirements

Supported Operating Systems:

• Windows 8.1 (x86 and x64)
• Windows 8 (x86 and x64)
• Windows 7 SP1 (x86 and x64)
• Windows Server 2012 R2 (x64)
• Windows Server 2012 (x64)
• Windows Server 2008 R2 SP1 (x64)

Hardware requirements:

• 1.6 GHz or faster processor
• 1 GB of RAM (1.5 GB if running on a virtual machine)
• 20 GB of available hard disk space
• 5400 RPM hard disk drive
• DirectX 9-capable video card that runs at 1024 x 768 or higher display resolution

Additional Requirements for the laptop:

• Internet Explorer 10
• KB2883200 (available through Windows Update) is required

And don't forget to reboot after updating your windows