Slowly...
In "Tools" > "Load package file (*.lpk)" go to the folder into which you extracted the "poweredby" download, open poweredby.lpk, click "Compile". Does it complete? There must be a green line "Compiled packed Poweredby 1.2. Success" in the message windows. Do no do "Use" > "Install" (unless you absolutely want the PoweredBy component on the component palette).
Now, still in the package editor dialog, go to the folder into which you extracted the "splashabout" download, open splashabout.lpk. Again: Click "Compile". Does ist complete? Click "Use" > "Install", but now do not confirm the question to rebuild the IDE and simply close the dialog. This way the splashabout component is marked for installation, but not yet installed.
Go to "Tools" > "Configure 'Build Lazarus'". In the "Cleanup" box, check "Clean all" and "Switch after building to automatically". Click "Build". This erases all compiled units and rebuilds the IDE from the start.
If this "clean build" does not resolve your issue there is something wrong with your installation. Which Lazarus version are you using? With Lazarus closed, you could rename the Lazarus config folder (*) which forces Lazarus to reset all settings to default - of course all your user defined settings will be lost and components will have to be reinstalled when you do this.
(*) You can determine the Lazarus config folder when you go to "View" > "IDE Internals" > "About ID" > "Global IDE options". Find the directory under "Primary config directory"
If this does not work it is probably easiest to reinstall Lazarus. Delete the config folder before the installation (or confirm to delete it when prompted during the installation).