The problem basically boils down to the fact that IE doesn't do auto-placement like every other browser.  Unless you tell it where you want each element within the grid, it will assume you want everything in column 1 row 1.  If this is a dynamic website you are better off with flexbox.  If its like your example and you know where each element is to go then you have to say so like so:
.grid {
  display: -ms-grid;
  display: grid;
  -ms-grid-columns: 50% 50%;
      grid-template-columns: 1fr 1fr;
}
.cell {
  border: 1px solid black;
}
.cell:first-of-type{
  -ms-grid-column: 1;
  -ms-grid-row: 1;
  -ms-grid-column-span: 1;
  }
.cell:last-of-type{
  -ms-grid-column: 2;
  -ms-grid-row: 1;
  -ms-grid-column-span: 1;
  }
<!DOCTYPE html>
<html>
<head>
  <meta charset="utf-8">
  <meta name="viewport" content="width=device-width">
  <title>JS Bin</title>
</head>
<body>
<div class="grid">
  <div class="cell">a</div>
  <div class="cell">     b</div>
</div>
</body>
</html>
 
 
All that being said if you really need IE support you should probably stick to flexbox.  They just do not get along.
UPDATE:
I recently stumbled upon an article outlining how Autoprefixer, if set up correctly, will do the grunt work for you if you are doing a grid-template-areas sort of thing. still no auto-placement but it does save you some time if you are making a template and are placing the elements anyway.
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