The code for the splash screen resides in the Default.aspx file which can be found in [INSTALLATION_DIRECTION]\bbappfx\vroot\browser\brandingpages\bbec\ on the web server.

For example, if you wanted to remove the Web Dashboards link and image you would remove the following lines from the file:

<td><div class="bigButton" id="webDashboardButton"></div></td>
<td>
                <h2><a id="webDashboardLink" href="#">Web dashboard</a></h2>
                <h3>Create visual overviews of your data using charts, gauges and other nifty widgets, and organize these views using pages and folders.</h3>
</td>

If you wanted to remove the Endpoints link you would remove the following

<td><div class="bigButton" id="developerButton"></div></td>
<td>
                <h2><a id="developerLink" href="#">Endpoints</a></h2>
                <h3>Browse the API's various endpoints and view their documentation.</h3>
</td>