This article cites the Brazilian environment ministry figure of 14,000 square kilometres of forest felled from Aug 05 to July 06. If a football field is 50x100m (0.005 square kilometres), that's roughly 2.8 million football fields at rate of ~5.32 per minute. I'd say their claim is a fair one, based on the round number provided by the ministry, my lack of a specific date range and the variability in size of football pitches.
That said, Brazil has made significant reductions in their clearing activities in recent years, and 6 football fields per minute is likely not the current rate. This 2010 Mongabay article claims that declines in annual worldwide totals for land clearing are primarily due to the reductions made by Brazil. (Emphasis below is mine)
FAO figures show deforestation across 121 tropical countries averaged 9.34 million hectares per year between 2000 and 2010, down from 11.33 million hectares per year in the 1990s. The decline has accelerated since 2005 due Brazil's dramatic reduction of deforestation in the Brazilian Amazon, which peaked at 27,772 square kilometers in 2004 but is expected to come in at less than 8,000 for 2010.