Welcome aboard!
Ai is a native file format of illustrator. Adobe produces and sells the program, hence it's not open source, nor is the file format -proprietary being the jargon.
Adobe created postscript format, which is used by the priners. Along .ps, came .eps and .pdf to share files in an easier manner with printers.
Illustrator's .ai format is the "twin brother" of pdf-s. They target .ai for editing, and pdf for sharing. For that purpose, pdf is standardised and the standards are available for other vendors. Inkscape uses ghostscript to open pdf files and cairo for saving as pdf.
Generally speaking, pdf is not an editing format and is not what inkscape was designed in first place -however still a very good free choice to use.
Illustrator had "generalized" their .ai format somewhere around illustrator 7 (?-can't recall, I haven't used illustrator.), and by saving to .ai format, a pdf equivalent is embedded in the document, for "just in case". At least, that's the default settings.
If you try to open an .ai file in inkscape, only the pdf part is taken into account (by third party ghostscript). The proprietary data is converted to a code only illustrator can understand. There is no specification given out for that by adobe.
So it seems your file was either drawn in a ten year old copy of illustrator or the pdf data was discarded from the file saving on purpose as far as I understand. While it being an "edit" format, assuming the designer you wanted to work with it later, he gave you only the "necessary" part. With smaller file size and unfortunately without the only part you could open without illustrator.
So either contact the author for a format you can use, or try to get someone saving your ai file to a reasonable format.
That would be svg the best if you want to use inkscape on it -which was developed with browser orientation, open and which is xml based.
Illustrator can save to svg as well, only a few parts are out the svg specs -like, cmyk support, gradient meshes and maybe some blend modes; also filters can end up rasterized in the process.
As it is a logo, probably it could be saved to svg without any trouble.
Worst case scenario, get an illustrator trial version for this 1 minute job.
