So you're using <Summary view>:

To get full descriptions, just select "All items" view from webpart properties:

, and add the Body column using "Modify view" ribbon button:

However, if it doesn't satisfy your needs (because "All items" view looks like a table), you will need some additional XSLT coding.
Creating analogue for summary view with uncut Body field
First of all, ensure that Announcements list is currently displayed in "table" mode, and Body and Created by fields are added into the view.
Open the page in SharePoint designer, switch to Code tab, and add <Xsl> tag to the XsltListViewWebPart markup.
Finally, paste the following code to the <Xsl> tag:
<xsl:stylesheet xmlns:x="http://www.w3.org/2001/XMLSchema" xmlns:d="http://schemas.microsoft.com/sharepoint/dsp" version="1.0" exclude-result-prefixes="xsl msxsl ddwrt" xmlns:ddwrt="http://schemas.microsoft.com/WebParts/v2/DataView/runtime" xmlns:asp="http://schemas.microsoft.com/ASPNET/20" xmlns:__designer="http://schemas.microsoft.com/WebParts/v2/DataView/designer" xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:msxsl="urn:schemas-microsoft-com:xslt" xmlns:SharePoint="Microsoft.SharePoint.WebControls" xmlns:ddwrt2="urn:frontpage:internal" xmlns:o="urn:schemas-microsoft-com:office:office">
<xsl:include href="/_layouts/xsl/main.xsl"/>
<xsl:include href="/_layouts/xsl/internal.xsl"/>
<xsl:template match="/">
<xsl:choose>
<xsl:when test="$RenderCTXOnly='True'">
<xsl:call-template name="CTXGeneration"/>
</xsl:when>
<xsl:when test="($ManualRefresh = 'True')">
<xsl:call-template name="AjaxWrapper" />
</xsl:when>
<xsl:otherwise>
<xsl:apply-templates mode="RootTemplate" select="$XmlDefinition">
<xsl:with-param name="ShowSelectAllCheckbox" select="false"/>
</xsl:apply-templates>
</xsl:otherwise>
</xsl:choose>
</xsl:template>
<xsl:template mode="header" match="FieldRef" />
<xsl:template mode="Item" match="Row">
<xsl:param name="Fields" select="."/>
<xsl:param name="Collapse" select="."/>
<xsl:param name="Position" select="1" />
<xsl:param name="Last" select="1" />
<xsl:variable name="thisNode" select="."/>
<tr>
<td width="80%" class="ms-vb" style="padding-bottom: 3px">
<span class="ms-announcementtitle">
<a onfocus="OnLink(this)"
href="{$FORM_DISPLAY}&ID={$thisNode/@ID}"
onclick="GoToLink(this);return false;" target="_self">
<xsl:value-of select="$thisNode/@Title"/>
</a>
<xsl:if test="$thisNode/@Created_x0020_Date.ifnew='1'">
<xsl:call-template name="NewGif">
<xsl:with-param name="thisNode" select="$thisNode"/>
</xsl:call-template>
</xsl:if>
<xsl:call-template name="FieldRef_Attachments_body">
<xsl:with-param name="thisNode" select="$thisNode"/>
</xsl:call-template>
<xsl:text disable-output-escaping="yes" ddwrt:nbsp-preserve="yes">&nbsp;</xsl:text>
</span>
<br />
<xsl:value-of select="'by'"/>
<xsl:text disable-output-escaping="yes" ddwrt:nbsp-preserve="yes">&nbsp;</xsl:text>
<xsl:value-of disable-output-escaping="yes" select="$thisNode/@Author.span"/>
</td>
<td width="20%" align="right" nowrap="nowrap" class="ms-vb">
<xsl:text disable-output-escaping="yes" ddwrt:nbsp-preserve="yes">&nbsp;</xsl:text>
<xsl:value-of select="$thisNode/@Modified"/>
</td>
</tr>
<tr>
<td colspan="2" class="ms-vb">
<div>
<xsl:value-of select="$thisNode/@Body" disable-output-escaping ="yes"/>
</div>
<!-- output the remaining fields -->
<xsl:for-each select="$Fields[not(@Name='LinkTitle' or @Name='Modified' or @Name='Author' or @Name='Body' or @Name='Attachments')]">
<br />
<xsl:value-of select="@DisplayName"/>:
<xsl:apply-templates select="." mode="PrintFieldWithDisplayFormLink">
<xsl:with-param name="thisNode" select="$thisNode"/>
</xsl:apply-templates>
</xsl:for-each>
</td>
</tr>
<tr>
<td colspan="2">
<font size="1">
<xsl:text disable-output-escaping="yes" ddwrt:nbsp-preserve="yes">&nbsp;</xsl:text>
</font>
</td>
</tr>
</xsl:template>
</xsl:stylesheet>
You will get exactly the same view as Summary view, but with uncut description.
Looks a bit complicated, huh? :) Actually mostly it's a copy-paste from OOTB SharePoint vwstyles.xsl file, with some pointed corrections. Creating this XSLT only took about 10 minutes.