I want to attach a statement explaining some things related to my tax return. Is there any way to attach a statement of explanation while submitting a tax return electronically using Free Fillable Forms? How?
Depending on what you need to explain, you can submit your electronic return without the supplemental information and subsequently mail a Form 8453 with the additional information. This is helpful for form 8489, for example, where you need to list every transaction reported by your stock broker on a 1099-B. See https://www.irs.gov/pub/irs-pdf/f8453.pdf for more details on this form.
If the information you need to submit an attachment for doesn't follow one of the options on that form, you will likely need to file a paper return or use a paid tax preparation service/application.
Limitations of FreeFile are explained here, along with a list of forms that are available: https://www.irs.gov/uac/List-of-Available-Free-File-Fillable-Forms
The "Attaching Statements" and "Write-in information" sections seem like they might apply to your situation.
Attaching Statements - If you need to add statements and you can't use Form 8453, U.S. Individual Income Tax Transmittal for an IRS e-file Return, to mail that information, you will not be able to use this program to efile your return
Identity Protection PIN's (IP PIN) - This program only supports the entry of a Primary taxpayer's IP PIN. If the spouse or dependents have an IP PIN, you cannot use this program to efile the return.
Writing In Information - Your ability to "write in" additional information to explain an entry is generally limited to the 1040 forms and some of the more frequently submitted forms. If you need to write in additional information on a form, other than the 1040 series, you may not be able to use this program to efile your tax return.
E-filing Forms - To efile forms, (except Form 4868) they must be attached to a 1040 series form (1040, 1040A or 1040EZ).
Form Limitations - There may be Known Limitations of forms you plan to complete. Please review them. A form limitation may keep you from completing or e-filing your return.