In your payment method you can assign an Info form block with setting (I took the Purchaseorder method as an example):
protected $_infoBlockType = 'payment/info_purchaseorder';
In this block you can assign the template file you want to use (see Mage_Payment_Block_Info_Purchaseorder for complete code):
protected function _construct()
{
parent::_construct();
$this->setTemplate('payment/info/purchaseorder.phtml');
}
And in this template file you can add info you want to dispay (the fields you added during the checkout):
<ul class="form-list" id="payment_form_<?php echo $this->getMethodCode() ?>" style="display:none;">
<li>
<label for="po_number" class="required"><em>*</em><?php echo $this->__('Purchase Order Number') ?></label>
<div class="input-box">
<input type="text" id="po_number" name="payment[po_number]" title="<?php echo $this->__('Purchase Order Number') ?>" class="input-text required-entry" value="<?php echo $this->escapeHtml($this->getInfoData('po_number')) ?>" />
</div>
</li>
</ul>
This will display the information entered at checkout in the order detail page.