How Do I Add HTML Support to the Additional Category Description Field?

sanitize_text_field() is intended to sanitize a value for use as plain-text. HTML tags are stripped in the process.

wp_kses() may be a more appropriate sanitization helper for your use-case and can be passed a custom list of permitted HTML tags. Alternately, you can pass a context name to use a pre-defined set. To use the list which is used in the process of sanitizing post content, for instance:

update_term_meta( $term_id, $key, wp_kses( $value, 'post' ) );

See the Theme Security section of the Theme Handbook for more information on Data Sanitization/Escaping in WordPress.