How can I prevent HTML tags getting rendered as plain text?
You need to unescape HTML. There are two working ways to accomplish this.
First you can mark your field as "raw" and the html would render correctly. 
{{ your_field | raw }} should render ok. 

As a second resort you can use twig's autoescape functionality:

{% autoescape false %}
{{ your_field }}
{% endautoescape %}


By this way you are telling twig not to escape html markup.