Remember, on ANY button click, or ANYTHING that causes a page post back?
Your on-load event fires EACH time, and runs BEFORE any other controls event stub.
So, be is a simple button and a click, or a dropdown list (say with autopost-back = true)?
The page load event will run first, and run EACH time, and BEFORE your event stub you have for the button or control in question.
As a result, the last 200 web pages I have created?
Every SINGLE one of those pages has this code stub:

if (!IsPostBack) 
 {
     // set controls and values - FIRST page load only!!!
 }

So, in your case, your page load event needs to be:

protected void Page_Load(object sender, EventArgs e)
    {
        if (!IsPostBack)
        {
            if (isMammals)
                mammals.Checked = true;
            if (isBirds)
                birds.Checked = true;
            if (isReptiles)
                reptiles.Checked = true;
            if (isAmphibians)
                amphibians.Checked = true;
            if (isFishes)
                fishes.Checked = true;
            if (isInvertebrates)
                invertebrates.Checked = true;
        }
    }

so, the event order is:

page load event

 your button event.

You can check further information about the Page events in the ASP.NET Page Life Cycle Overview
So, after the page load code runs, then you have this code + your button code.

isMammals = mammals.Checked;
isBirds = birds.Checked;
isReptiles = reptiles.Checked;
isAmphibians = amphibians.Checked;
isFishes = fishes.Checked;
isInvertebrates = invertebrates.Checked;

So, above code, when it runs will run AFTER the page load code that sets the check boxes. If you leave out the !IsPostBack, then the page load event will change and OVERWRITE any changes the user makes.
As a result, you really can't even make a practial working web page UNLESS you include + use + adopt + take into account the required !IsPostBack stub that you need in 99% of cases.
So, in page load event, you are 100% free to load up a gridview, load up some dropdown lists, setiup some checkboxes (as your example has), but since this code runs on EVERY page-post back?
Then you need that code stub inside of the !IsPostBack stub, and without that code wrapped inside of the !IsPostBack code block, then you are overwriting the users choice each and EVERY time the page loads.